# Eigen-energies and eigenstates of a tri-atomic system

## Homework Statement

An extra electron is added to one atom of a tri-atomic molecule. The electron has equal probability to jump to either of the other two atoms.

(a) Find the eigen-energies for the system. Assume that the new electron energy ##\bar{E_{0}}## is close to the non-hopping case energy ##E_{0}##. Draw an energy level diagram.

(b) Find one normalized eigenstate for the system.

## The Attempt at a Solution

(a) The only information available to me are that the the electron has equal probability to jump to either of the other two atoms. and the system has three atoms. How can these two pieces of information be used to find the eigen-energies of the system? Am I missing something?

## Answers and Replies

DrDu
Science Advisor
I suppose you can assume that the three atoms are identical.

How might that help me? For now, all I can say is that the electron has the same eigen-energy for being in any of the three atoms i.e. in any one of the three states.

Is that all I can say for the eigen-energies of the system?

DrDu
Science Advisor
I would also say that the model is somewhat ill specified. Did you do define "hopping" in class, maybe in connection with the Hubbard or Hueckel model? I think what you are supposed to assume is that the electron can be in one specific orbital on the atom, which is identical for all 3 atoms. Equal hopping probability translates into equal hamiltonian and overlap matrix elements between the orbitals. You then can set up some 3x3 Hamiltonian and Overlap matrix and try to solve it.

I think I will have to refer back to my QM textbooks and learn more about related topics before I can tackle this problem.

Can you please mention the topics that I must learn and be familiar with before I can answer this question?